Finite fields are fields containing only a finite number of elements, and linear equations over finite fields are equations of which each factor is an element in finite fields. Finding solutions to linear equations over finite fields is widely used in various engineering fields, such as the field of cryptography, and also the field of solving other mathematical problems.
Methods adopted in solving linear equations include Gaussian elimination method and Gauss-Jordan method. By Gaussian elimination method, linear equations are multi-iterated into upper or lower triangular forms, wherein each iteration operation includes three sub-operations: finding pivot, normalization and elimination. If the equations are solvable, then the final solutions are obtained by substitution operation. Gauss-Jordan method, a variant of Gaussian elimination method, is able to solve linear equations by multiple iterations, but it consumes more resources than Gaussian elimination method.
Solving linear equations is a highly computationally complex and time-consuming issue. At present, there is still large room for optimization of solving linear equations, especially for optimization of those over finite fields. So far, devices dedicated to solving linear equations over finite fields have not yet been reported.